Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes

Abstract

Target shape, not just size, plays a pivotal role in determining detectability during random search. We analyze intermittent Lévy walks in three dimensions, and mathematically prove that the widely observed Cauchy strategy (Lévy exponent $μ= 2$) uniquely achieves scale-invariant, near-optimal detection across a broad spectrum of target sizes and shapes. In a domain of volume $n$ with boundary conditions, expected detection time for a convex target of surface area $Δ$ optimally scales as $n/Δ$. Conversely, Lévy strategies with $μ< 2$ are slow at detecting targets with large surface area-to-volume ratios, while those with $μ> 2$ excel at finding large elongated shapes but degrade as targets become wider. Our results further indicate a continuous geometric transition: volume dictates detection near $μ= 1$, ceding dominance to surface area as $μ\to 2$, after which surface area and elongation couple to govern detection. Ultimately, 3D search introduces a pronounced sensitivity to target shape that is absent in lower dimensions. Our work provides a rigorous foundation for the Lévy flight foraging hypothesis in 3D by establishing the scale-invariant optimality of the Cauchy walk. Furthermore, our results reveal dimensionality-driven shape vulnerabilities and offer testable predictions for biological and engineered systems.

Figures (4)

• Figure 1: Search efficiency across Lévy regimes and target geometries. Summary of the performance of ballistic ($\mu < 2$), Cauchy ($\mu = 2$), and diffusive ($\mu > 2$) intermittent Lévy walks across three fundamental target geometries—balls, discs, and lines—at both small and large scales. Performance is measured relative to the optimal search strategy tuned to the specific geometry. In our rigorous proofs, efficiency ($\checkmark$) indicates that the performance ratio is tight up to lower-order terms, and inefficiency ($\times$) corresponds to a polynomial gap in system parameters. Green and Red denote efficiency and inefficiency, respectively, as established by rigorous mathematical proofs (and corroborated by simulations); Yellow indicates efficiency supported solely by numerical simulations.
• Figure 2: Intermittent search, Bounding box, and Universal lower bound construction. Fig. \ref{['fig:1patch']} provides a visualization of an agent performing an intermittent Lévy walk (for simplicity of presentation, it is two-dimensional). The points to be detected (e.g., food patches) are represented by red circles, and the surrounding grey area indicates the detectable region, which we refer to as the target $S$; similarly, the blue area around the searcher represents its own detection radius. In this specific instance, the agent successfully detects the prey via this range. Notably, because the search is intermittent, the agent cannot detect targets while in transit; detection occurs only upon stopping at the locations indicated by the black dots. Fig. \ref{['fig:1delta']} shows a target (red shape on the left) together with its minimum surface area bounding box $B = \texttt{Box}(S)$ (grey box on the left), which is not necessarily axis-aligned. The panel on the right illustrates the projection of the shape onto the largest face of the box; In this example, $\Delta_P$ equals the red area, while $\Delta_B$ equals the red area plus the grey area. Figure \ref{['fig:1tiling']} depicts the tiling argument used to derive the universal lower bound in Eq. \ref{['universal-lower']}. For clarity, the boxes are shown as axis-aligned; in the general case, however, they need not be aligned with the coordinate axes.
• Figure 3: Comparing the detection times of different Lévy walks. All simulations correspond to the 3D torus $\mathbb{T}_n$ with volume $n=512^3$. Fig. \ref{['fig:1upper_bound']} presents a comparison of the detection times of the Cauchy walk for Line, Disk, and Ball targets, all sharing the same projected surface area $\Delta_P$, alongside the corresponding universal lower bound (dashed line). Fig. \ref{['fig:2relative_detection_time_ball']} displays detection-time ratios normalized by the detection time of the Cauchy walk ($\mu = 2$) for ball-shaped targets with surface area $\Delta$ (color-coded), emphasizing the relative performance of different exponents. Figs. \ref{['fig:2relative_detection_time_disk']} and \ref{['fig:2relative_detection_time_line']} display analogous results for a disk and line-shaped targets, using the same experimental setup. Takeaway: $\mu = 2$ stays near baseline across geometries; $\mu \neq 2$ exhibits geometry-dependent slowdowns.
• Figure 4: Detection-time phase transition induced by varying $\mu$ and dependency on elongation. All simulations are performed on a three-dimensional torus $\mathbb{T}_n$ with volume $n = 512^3$. Fig. \ref{['fig:3detection_time_ratio_volume']} shows the ratio of detection times for a ball and a line of equal volume, for several values of $\mu$. The $x$-axis corresponds to the line length $D_{\mathrm{Line}}$, while the diameter of the corresponding ball is chosen so that the two volumes are equal. The dashed grey line shows the ratio of the surfaces of the two targets, representing a theoretical estimation for the detection time ratio of the Cauchy walk. Similarly, Fig. \ref{['fig:3detection_time_ratio_surface']} presents the ratio of detection times for a ball and a line with the same surface area $\Delta$, shown on the $x$-axis. In this figure, the dashed line represents the ratio of the volumes of the two targets, serving as a theoretical estimation for the Lévy walk as $\mu \to 1$. Fig. \ref{['fig:3varying_delta_small_mu']} presents detection times for Lévy walks with $\mu \in \{1, 1.5, 2\}$, where the targets are rectangles with side lengths $\Delta^\delta$ and $\Delta^{1-\delta}$. Similarly, Fig. \ref{['fig:3varying_delta']} shows the result for the same set of experiments when $\mu \in \{2, 2.5, 3\}$. Takeaway: the governing geometric control shifts from volume to surface, and then to surface and elongation as $\mu$ increases.

Theorems & Definitions (48)

• Claim 2.1
• Claim 2.2
• Corollary 2.3: Monotonicity
• proof
• Theorem 2.4
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• proof